Taylor polynomial

The degree- $n$ polynomial matching $f$ and its first $n$ derivatives at $x=a$ — the best local polynomial approximation.

The Taylor polynomial of degree $n$ for $f$ centered at $a$ is $P_n(x)=\sum_{k=0}^{n}\dfrac{f^{(k)}(a)}{k!}(x-a)^k$ — constructed so $P_n$ and $f$ share the same value AND the same first $n$ derivatives at $x=a$ . Increasing $n$ generally makes $P_n$ hug $f$ 's graph more closely over a wider range near $a$ .

Examples

The degree- $2$ Taylor polynomial for $e^x$ at $a=0$ is $P_2(x)=1+x+\dfrac{x^2}{2}$ .